The Whitney method of fundamental solutions with Lusin wavelets

Abstract

We establish the theoretical foundation for a variant of the method of fundamental solutions (MFS), where the source points $\{q_j\}_{j=1}^\infty$ accumulate towards the domain in a Whitney fashion, meaning that their separation is proportional to the distance to the domain. We prove that the normalized Lusin wavelets $ψ_j(w) = b_j(w-q_j)^{-2}$ constitute a generalized basis, known as a frame, for the Hardy subspace of $L_2$-traces of holomorphic functions on the domain. Consequently, our method, where $ψ_j$ are used as basis functions in the MFS, enables a numerically stable approximation of solutions to Laplace boundary value problems, even when the solutions lack analytic continuation across the boundary. Despite the source points accumulating towards the domain, our computations achieve at least 12 digits of accuracy uniformly up to the boundary, including cases when the solution lacks analytic continuation or when the boundary has corners.